A graph has a representation modulo n if there exists an injective map f: {V(G)} -> {0, 1, ... , n @ 1} such that vertices u and v are adjacent if and only if |f(u)@f(v)| is relatively prime to n. The representation number is the smallest n such that G has a representation modulo n. We seek the maximum value for the representation number over graphs of a fixed order. Erdos and Evans provided an upper bound in their proof that every finite graph can be represented modulo some positive integer. In this note we improve this bound and show that the new bound is best possible (Refer to PDF file for exact formulas).

Publication Date



Copyright 2003 The Author

The author is indebted to a referee for many valuable suggestions, including an improved presentation of the proof of Theorem 3. This work was partially supported by a RIT College of Science Dean's Summer Research Fellowship Grant and a Reidler Foundation Grant. Thanks to Anthony Evans and Garth Isaak for useful discussions. Finally the author thanks Dustin Sheffield, now an undergraduate at Virginia Tech, for preliminary computer simulations.

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Department, Program, or Center

School of Mathematical Sciences (COS)


RIT – Main Campus